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Quantum Mechanical Simulations on a Beowulf Cluster Nathan Shupe Student Investigator Swarthmore College Dr. J. Anthony Gualtieri Mentor Code 935 V ISITING S TUDENT E NRICHMENT P ROGRAM VSEP

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Abstract The objective of this project was to develop a parallel simulation code from a similar single processor code, and then run the parallel simulation code on a Beowulf cluster machine. The code studied utilizes a random path model using entwined paths that can be shown to model the quantum mechanical properties of a particle in a box. By running the code on Medusa, a Beowulf cluster machine environment, and confirming our results from simulations on a single processor, it has been shown that this parallel environment is an effective tool for computations which simulate quantum mechanical systems. In addition, significant progress has been made in showing that the equations of quantum mechanics are governed by an underlying stochastic process.

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Outline of Presentation I.Introduction A.Quantum History B.Feyman Paths C.Entwined Paths D.Importance of Research II.Methods A.Benefit of Parallel Simulation B. Beowulf Cluster C.Simulation Code Development III.Results A.Dirac Propagator for a Free Particle B.Time evolution of the Ground State of a Particle in a Box. IV.Discussion A.Space-time History B.Dirac Equation C.Periodic Boundary Conditions V.Conclusion A.Accomplishments B.Future Work VII.References / Acknowledgements

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Quantum History I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments Conventional quantum mechanics has the wave-function which allows us to decode information about nature. In this respect, quantum mechanics is well- defined in that we can precisely predict information about physical systems. The question remains, however, of how nature can encode the information in the wave- function. Conventionally, the answer to this question has been we do not and cannot know.

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Quantum History = The wave-function 1 is simply an algorithm which describes our knowledge of the universe. = (or) = The wave-function corresponds to something physical in reality. Quantum Mechanics is an epistemological model. (or) Quantum Mechanics is an ontological model. The choice at hand: I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References Acknowledge- ments 1 We shall define the wave-function as the probability amplitude of finding a particle at a certain position and time.

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Feynman Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments The Problem: At initial time a particle starts from the point and moves to a final destination at time. The probability amplitude, or kernel, to get from to is given by: where is the kernel, is a path, and is the probability amplitude of that path.

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Feynman Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments Thus, according to Feynman’s formulation, the probability amplitude or kernel of a particle moving from one point to another in space-time is the sum of all of the amplitudes of all of the possible paths from the initial position to the final destination. One way to define paths between points in space-time is to apply a lattice structure to space-time. In order to consider relativistic particles, we will use the chessboard or checkerboard model.

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Feynman Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments x t If the time and space axes are in the right units 2, then relativistic particles will move on diagonals, similar to the moves of a bishop. FIG. 1: Chessboard Model. An example of a space- time trajectory of a relativistic particle is shown in Fig. 2. Fig. 2: Path of a Relativistic Particle from a to b. 2 The velocity of light, the mass of the particle, and Planck’s constant are all unity. x t a b

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Feynman Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments If we assign ε to be the magnitude of time steps on the lattice, and that the path direction can only change at the boundaries of the time steps, then the probability amplitude of traversing a path is given by: where is the number of corners, or direction changes of the path, and is. This definition of the path probability amplitude changes the expression of the kernel.

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Feynman Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments The kernel can be calculated by adding up the contributions for the paths with corners. is the number of paths possible with corners. As a result of our chessboard formulation, the probability amplitude and kernel are now valid for a free particle moving at relativistic speeds in one dimension. Therefore, the kernel should be equivalent to the Dirac propagator.

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Entwined Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments Recall that relativistic particles move on diagonals in the chessboard model, similar to the moves of a bishop. Note that once a bishop moves from its initial position, it can always return to that position, provided that there are no other pieces in the way. As a result, a single bishop could trace out every possible path on the chessboard from its initial square to another, simply by moving forward to the desired square, returning to its original, and then repeating the process for a different path.

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Entwined Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments x x t Feynman paths restrict the moves of the bishop to a forward direction… FIG. 4: Chessboard Model for a Feynman Path.

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Entwined Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments x whereas entwined paths permit backward moves as well. FIG. 5: Chessboard Model for an Entwined Path. t

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Entwined Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments For entwined paths, a blue line denotes a path moving forward in time (electron), and a red line denotes a path moving backward in time (positron). The path is constructed as follows: x t electron positron interaction FIG. 6: Entwined Path Construction.

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Entwined Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments The simulation code generates entwined paths somewhat differently, since half of the entwined path holds all of the information necessary to generate the entire path. x t FIG. 7: Right Envelope of an Entwined Path.

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Entwined Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments Feynman paths have a many-to-one correspondence with the particle they are describing (i.e., many paths must be generated in order to complete the ensemble and extract the propagator). Entwined paths, on the other hand, have a one- to-one correspondence with the particles they describe, since our method of constructing entwined paths starting and ending at the same point in space-time creates a single path. Therefore, entwined paths can more easily be associated with physical trajectories.

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Entwined Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments Another advantage of entwined paths is that they create the phase of the wave- function without invoking an analytic continuation. That is, the phase of the wave-function does not have to be artificially created by making the probability amplitudes complex, but rather comes about naturally as the result of the space-time geometries of the entwined paths.

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Entwined Paths I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments Essentially, with entwined paths we are trying to do for quantum mechanics what Weiner paths did for diffusion. That is, we are trying to show that these paths actually correspond to physical trajectories, and in so doing show that the mechanics of propagation and the measurement postulates are the result of an underlying stochastic process 3. 3 A process governed by probabilities, like the flipping of a coin.

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Importance of Research I. Introduction A.Quantum History B.Feynman Paths C.Entwined Paths D.Import. of Research II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments By running the code on Medusa 4, and confirming our results from simulations on a single processor, we have shown that this parallel environment is an effective tool for computations which simulate simple quantum mechanical systems. A model of this kind may lead to further insights into quantum mechanics, thereby allowing more effective application of our understanding of quantum mechanics (e.g., quantum computing.) 4 A 64 node Beowulf cluster machine. The nodes have 2 processors (AMD Athlon 1.2 GHz, 512K of cache) each and are connected by a MYRINET (Myricom) network.

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Benefit of Parallel Simulation I. Introduction II. Methods A.Benefit of Parallel Simula- tion B.Beowulf Cluster C.Simula- tion Code Develop- ment III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments Since our model is based on an underlying stochastic process, simulations with better statistics will produce better results. One way to increase the statistics of a simulation is to run the code on many processors at once, such as in a parallel computing environment, and then compile the results from all of the processors in order to produce a result. For this project, we used Medusa, a Beowulf cluster machine environment.

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Beowulf Cluster I. Introduction II. Methods A.Benefit of Parallel Simula- tion B.Beowulf Cluster C.Simula- tion Code Develop- ment III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments A Beowulf Cluster, having a distributed memory system, is well suited to run many copies of the simulation code simultaneously. Essentially, we ran identical copies of the code on all of the processors, with different parameters (origin, probability of state change) for each processor. Thus, each processor simulated a single particle in a box. After the simulations had completed, we recompiled the information into an output file on a single processor machine, and used a graphing utility to visualize the results.

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Simulation Code Development I. Introduction II. Methods A.Benefit of Parallel Simula- tion B.Beowulf Cluster C.Simula- tion Code Develop- ment III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments Initially, the simulation code was developed in a single processor environment, which allowed for easier coding and testing. Once satisfied with the single processor code, we ported the code to a parallel environment in order to produce better statistics and to more vigorously test the validity of our model. FIG. 8: Entwined Path in a Box.

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Simulation Code Development I. Introduction II. Methods A.Benefit of Parallel Simula- tion B.Beowulf Cluster C.Simula- tion Code Develop- ment III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments The criteria of our code assessment were: (1) the code successfully stored the projection of each particle’s space-time history and when necessary referenced this history to produce a state change, and (2) that if given initial conditions for the energy eigenfunctions, the eigenfunctions of the particle were shown to persist.

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I. Introduction II. Methods A.Benefit of Parallel Simula- tion B.Beowulf Cluster C.Simula- tion Code Develop- ment III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments Simulation Code Development The algorithm that generates an entwined path in a box [1] was originally written in MATLAB (MathWorks, version 6.5) which meant that before the code could be ported to a parallel environment, it would have to rewritten in another language, such as C or Fortran (we chose to rewrite in C). At first, we tried to do this manually, but later elected to instead use the MATLAB C Compiler. FIG. 9: Simulation Code Development.

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Dirac Propagator for a Free Particle I. Introduction II. Methods III.Results A.Dirac Propa- gator for a Free Particle B.Time Evo- lution of the Ground State of a Particle in a Box IV.Discussion V.Conclusion VI.References / Acknowledge- ments FIG. 10: Dirac Propagator for a Free Particle. This result came from a simulation code which did not retain the space-time history, and was run on Medusa, a Beowulf cluster.

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Time Evolution of the Ground State of a Particle in a Box I. Introduction II. Methods III.Results A.Dirac Propa- gator for a Free Particle B.Time Evo- lution of the Ground State of a Particle in a Box IV.Discussion V.Conclusion VI.Acknowledge- ments / References FIG. 11: Time evolution of the ground state of a particle in a box. This result came from a simulation code which did not retain the space-time history, and was run on a single processor machine.

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Space-time History I. Introduction II. Methods III.Results IV.Discussion A.Space- time History B.Dirac Equation C.Periodic Boundary Condi- tions V.Conclusion VI.References / Acknowledge- ments Thus far, our results are limited to MATLAB simulate code that does not retain a history of past events for each point in the space-time grid. The newest versions of the simulation code do meet this criterion, which is in place to accelerate the convergence of the code (from to, where is the number of time steps). Preserving the space-time history speeds up convergence because particles can look at the eigenspace and minimize fluctuations if necessary.

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Dirac Equation I. Introduction II. Methods III.Results IV.Discussion A.Space- time History B.Dirac Equation C.Periodic Boundary Condi- tions V.Conclusion VI.References / Acknowledge- ments In Fig. 11, notice that the wave-function has jagged edges at the boundaries of the box, and becomes worse as time progresses. Initially, we thought these rough edges were a problem with the code, but recently found that the problem arises from a property common to all discrete models of the Dirac equation for a particle in the box. Given an initial trigonometric condition for the wave-function, the Dirac equation is not separable.

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Periodic Boundary Conditions I. Introduction II. Methods III.Results IV.Discussion A.Space- time History B.Dirac Equation C.Periodic Boundary Condi- tions V.Conclusion VI.References / Acknowledge- ments FIG. 12: Entwined path in a box with periodic boundary conditions. One way to avoid the problems with the Dirac equation for a box is to set up periodic boundary conditions. Fig. 12 shows the output of the newest version of the simulation code. Periodic boundary conditions allow the Dirac equation to separate given trigonometric initial conditions.

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Accomplishments I. Introduction II. Methods III.Results IV.Discussion V.Conclusion A.Accomp- lishments B.Future Work VI.References / Acknowledge- ments The entwined path model preserves the propagator for the particle in a box and retains the energy eigenfunctions given initial conditions. We have shown that running the simulation code on a Beowulf cluster is a simple and efficient way to get better statistics for the simulation, and have therefore demonstrated the effectiveness for simulating quantum mechanical systems of this kind using this type of parallel environment.

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Future Work I. Introduction II. Methods III.Results IV.Discussion V.Conclusion A.Accomp- lishments B.Future Work VI.References / Acknowledge- ments In the near future we hope to show that the Born Postulate 5 holds under a simple measurement scheme for our simulation of a particle in a box. Also, we plan to rewrite the simulation code so that both the right and left envelope of the entwined paths check the eigenspace in order to determine state change. Lastly, we also plan to further parallelize the code so that processors can share their space- time histories. 5 The postulate states that the probability of measurement of a particle in a certain state is the square of the modulus of the wave-function.

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References I. Introduction II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments [1]R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw- Hill, New York, 1965). [2]G. N. Ord. box-path-generate.m, 2003. [3]G. N. Ord and J. A. Gualtieri. The Feynman Propagator from a Single Path. Physical Review Letters, 89(25):1-4, 2002. [4]G. N. Ord and R. B. Mann. Entwined paths, Difference Equations, and the Dirac Equation. Preprint, 2002.

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Acknowledgements I. Introduction II. Methods III.Results IV.Discussion V.Conclusion VI.References / Acknowledge- ments Dr. Marilyn Mack VSEP Director Marci Delaney VSEP Coordinator Joan Puig Student Dr. J. A. Gualtieri Mentor, Code 935 Dr. G. N. Ord Collaborator, M.P.C.S., Ryerson University Nathan Redding Student Thank you.

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